In the sequel to this article we carefully study the relationship between this construction and what we have done in this article. The gist of this story is that, for a general universal stability condition \(\phi \in V_{g,n}\) , the cone stack \(\widetilde{\mathcal {J}}^{trop}_{g,n}(\phi )\)  defines a proper subdivision of the universal tropical Picard variety \(\operatorname{\mathcal {T}ro\mathcal {P}ic}_{g,n,d}\) . This induces a logarithmic modification \(\mathcal {J}_{g,n}^{log}(\phi )\mathbb {G}_m \rightarrow \operatorname{\mathcal {L}og\mathcal {P}ic}_{g,n,d}\)  – effectively a blowup – by base change along the tropicalization map. As an application, this approach allows one to readily relate the spaces constructed here with Abel-Jacobi theory (see e.g. [1]} and [2]}). The perspective adopted in this paper and the sequel is for instance crucial in ongoing work of Holmes, Pandharipande, Pixton, Schmitt, and the second author regarding the double ramification cycle.
